197 - TWMS Journal of Applied and Engineering Mathematics

Transkript

TWMS J. App. Eng. Math. V.4, N.2, 2014, pp. 175-198.
"YARMAN-36 MAKAM TONE-SYSTEM" FOR TURKISH ART MUSIC
OZAN YARMAN
1,2
, M. KEMAL KARAOSMANOĞLU 3 , §
Abstract. This study offers a mathematically rigorous, yet straightforward, fixed-pitch
tuning strategy to the problem of adequate sounding and notating of essential Turkish
makam genera, in contradistinction to the praxis-mismatched music theory cast in effect
known as Arel-Ezgi-Uzdilek (AEU). It comprises 36 tones locatable just by ear, via
counting exact 0, 1 and 2 beats per second when listening to given octave, fifth and third
intervals, starting from an algebraically attained reference frequency for A at 438.41
Hertz, very near the international standard A=440 Hz. The so-named Yarman-36 makam
tone-system proposed in this paper accounts for hitherto omitted pitches in Uşşak, Saba,
Hüzzam, etc... at popular transpositions, each corresponding to a habitually used Ahenk
(concert pitch level specified by a chosen Ney reed), by virtue of being based on a
twelve-by-twelve triplex structure of exclusively tailored Modified Meantone Baroque
Temperaments. It thus also features pleasant shades of key-colors supporting polyphonic
endeavours in line with Western common-practice music.
Keywords: Arel-Ezgi-Uzdilek, makam (maqam), tone-system, tuning, temperament.
AMS Subject Classification: 00A65
1. Problems with the established 24-tone makam theories
Makam in Turkey, and homonymously elsewhere across the Middle East from Morocco
to Uyghur autonomous region of China, designates a musical mode, or a family of kindred modes, consisting of a set of “more or less” fluid pitches (called perdeler ), with distinctly embedded intonation (baskı) and inflexion (kaydırma, oynaklık ) attributes, the
entirety of which remains dependent on the classical rules of thematic flow (called seyir ).
[16,20,24,38,48,49,52,53,54,70]
Due to said ambiguous traits of Makam music, it is exceedingly difficult to pinpoint –
especially within the context of live performance – the precise microtones typically used
by a genus or scale. In addition, the reference frequency for any particular ensemble can
be selected from among no less than a dozen options, each bearing such habitual names
Musicology Department, İstanbul University State Conservatory, 81300 Kadıköy, İstanbul, Turkey.
Science & Art Counsellor to the Rectorate (till August 2014), Başkent University, Bağlıca 06810,
Ankara, Turkey.
e-mail: [email protected];
3
Art and Design Faculty, Yıldız Teknik University, Esenler 34220, İstanbul, Turkey.
e-mail: [email protected];
§ Manuscript received: 27 October 2013.
TWMS Journal of Applied and Engineering Mathematics, Vol.4, No.2; © Işık University, Department
of Mathematics 2014; all rights reserved.
1
2
175
176
TWMS J. APP. ENG. MATH. V.4, N.2, 2014
as Bolahenk, Süpürde, Mansur, etc..., any of which indicates a specific Ney Ahenk
corresponds to a chosen size of reed (Fig. 1).
i
that
Figure 1. Different sizes of Ney reeds corresponding to different Ahenks
(Fingering is preserved despite the change of dimensions, which results in a
key transposed instrument serving as the ensemble’s reference pitch).
Understandably, and owing also to the adverse influence of a bicentennial flurry of
Westernization in Turkey and beyond, there has been considerable efforts to accurately
identify or determine the total amount of perdeler that make up the master tuning grid
of Makam music. All the way from 17 through 24, 29, 36, 41, 48, 51, 53, 60, 65, 72 up to
79 unequal or equal divisions of the octave have thus far been variously endorsed and/or
applied in the literature. [1,4,19,21,22,23,28,32,33,36-38,43,50,53,55,56,59-64,66-69]
Howcome so many arithmetically interesting, yet basically irreconcilable, tuning schemes
contend side by side to explain makamlar today should not astound the reader. The latecomer majority of the above-mentioned tone-systems aspire to reinstate what the historically adopted Turkish/Arabic/Persian 24-tone music theory templates fundamentally neglect or evade: i.e., “unruly microtones” that significantly overshoot or undershoot 12-tone
Equal Temperament pitches at customary musical registers, transpositions, and modulations; which are pertinaciously executed by performers on their instruments, despite having
remained non-systematized heretofore to the agreement of the majority.
As it so happens, investigations have firmly ascertained of late that, the official tonesystem of Turkish Classical/Art music known as Arel-Ezgi-Uzdilek (AEU) does not at
all reliably reflect practice. It is demonstrably on account of the failure of this 24-tone
i
Bolahenk with perde rast (second partial blown from all fingerholes of the Ney closed) at D; Davud
with rast at E; Şah with rast at F; Mansur with rast at G; Kız with rast at A; Müstahsen with rast at B;
and Süpürde with rast at C. Observe, that perde rast can be made to correspond to any tone of Western
common-practice music, including all the half-tones in-between the naturals.
YARMAN ET AL.: "YARMAN-36 MAKAM TONE-SYSTEM" FOR TURKISH ART MUSIC
177
Pythagorean tuning ii (cf. Formula 1.1) [25,46,58] at embodying or expressing via staff
notation the multifarious neutral or middle second flavors iii [5,60] measured in audio recordings [2,8,10-15,26,30,31,34,35,44,45,49], and thus, inextricably peculiar to the genre.
In contrast, manifestly amiss in the quarter-tonal setup of Arab and Persian maqam
/dastgah treatises [19,23,50,51,53] is a mathematically complete (i.e. transposition-wise
fully navigable) model incorporating minute “commatic alterations” inherently found in
AEU, which are otherwise at the disposal of executants of native free-pitched instruments.
The situation can be just as perplexing in the Arabic and Persian world of traditional
music-making as it is in Turkey, due to each faction prioritizing an idiosyncratic body of
tuning criteria at the outset. The issue is markedly as complex as the methodical and clever
blending of the commatic soundscape with the quarter-tonal.
Let’s provide a direct analogy to shed more light on the complexity of the matter:
Whereas the utilization of 12-tone Equal Temperament or a sibling cyclic tuning iv [7,9,29,
47,57] on keyboard and fretted instruments in Western Classical/Contemporary music
is not in the least unacceptable – viz. one can interchangably perform (and even rearrange) a piece written for Trombone or Violin on a Piano or Guitar without grossly
misrepresenting or distorting the intended music (i.e., within the recognized boundaries of
instrumentalism), fretting the Tanbur or affixing mandallar v on a Kanun strictly according
to AEU will be disastrous for Turkish Art music performance.
ii
AEU is generated from an initial relative frequency (1~1) – dubbed perde kaba çargah and notated
) – by going up eleven pure fifths
as a ledger lined C (Do) below the first line of a G-cleffed staff (
(3~2) therefrom, then again up twelve pure fourths (4~3) once more therefrom, and bringing all resultant
ratios within the range of a single octave (2~1) via the required octave transpositions. “Going up” here
signifies multiplication of either the initial 3~2 or 4~3 ratio by itself to arrive at the next ratio, which is
again multiplied by same to yield a further ratio, etc... Transposing by the octave means that a fraction’s
even number numerator should be divided by 2 or even number denominator multiplied by 2 (should the
fractional value be greater than 2) until the ratio comes to reside between 1 and 2. The whole operation
can be mathematically expressed as follows in Formula 1.1:
Formula 1.1 (deriving the frequency ratios of Arel-Ezgi-Uzdilek )
¯
1 @ Rn
where
n
m
1~2

@ 2
0, 1, 2, 3, ...11
1, 2, 3, ...12
(1.1)
if R
if R
3~2
4~3
provided that,
m

16
0
if
if
¯
¯
>2
<2
The outcome of 24 distinct pitches (not including the octave, and ordinarily sounded at Bolahenk with
perde yegah or D=220 Hz) is thusly so-called Pythagorean due to the prime factorization of the numerators
and denominators in these ratios producing only 2’s and 3’s – which have been held as core mystical &
celestial numbers by the adherents of said ancient school of Pythagoras. (Accompanying endnotes)
iii
Characterizable by a spectrum of superparticular Just Intonation ratios that proceed as 11~10, 12~11,
13~12, and 14~13 within a given whole-tone range, which are altogether absent in AEU at indispensible
locations. (Accompanying endnotes)
iv
i.e., any of the countless finely calculated circulating Well-Temperaments or Modified Meantone
Temperaments found in the vast literature on the historical tuning of common-practice European music,
by which a chain of selectively sized perfect fifths wrap around to a full circle at the 12th step – resulting
in the return to the tone of origin as well as the ability to transpose unhindered, while maximizing aurally
favorable central tonalities and yielding various key-colors. (Accompanying endnotes)
v
Small metallic levers that are altered by the performer on-the-fly to modify the vibrating length of
string courses. Every mandal is affixed to the Kanuns in the construction phase, and the player does not
have the option to change their default positions.
178
So too is the case analogous with dividing the octave into 24 equidistant parts when
playing on a generalized Arabic Qanun or a Persian/Azeri Tar according to Formula 1.2:
Formula 1.2 (relative frequency "¯" and cent value "" of the quarter-tone interval )
¯
2 1~24
(1.2a)
º
24
(1.2b)
2
exp
1
ln 2
24
yielding 1200
(1.2c)
log 10 ¯
log 10 2
50 
or otherwise, 1200 log 2
or still, 1200
1
24
¯
(1.2d)
50 
50 
where the result is the quarter-tone step of 50 cents, twenty-four of which synthetically
added together total 1200 cents vi [27,40], hence, the octave.
Thus, neither the Turkish 24-tone Pythagorean, nor the Arabic/Persian quarter-tonal
templates (where twelve base pitches are either commatically or quarter-tonally etched from
the remaining twelve) can wholesomely house the intended music for dastgaha/makamlar/
maqamat/mughamlar [6] or suchlike modulations to particularly Saba, Uşşak, Hüseyni,
Hüzzam, Karcığar, Suzinak, Bayyati, Shur, Dashti, etc..., without causing a dilettante of
the genre to wince upon hearing them; since the aural margin of error can indeed be very
narrow for certain critical microtones during modal progression [13].
In other words, it is impossible to perform authentic music in such modes based on the
standardized 24 pitches to the octave systems of the diametrically opposed cultures of the
geography, without detuning the strings, adding or shifting frets as required, or employing
an ad hoc (e.g., unmethodical) mandal configuration.
Furthermore, the absence of neutral or middle second accidentals is as glaring in AEU
as the dearth of commatic nuances are in the notational symbology of Arabic and Persian music theory. A natural consequence of all this has been that, several alternative
tuning models have been proposed in the past decades – particularly in Turkey, with a
conscious aim to remedy the aforementioned shortcomings of the 24 pitches to the octave
methodologies, including some by the first author himself [59-63].
vi A unit of intervallic measure in the logarithmic scale, first proposed by Alexander J. Ellis in 1885 in
his revised translation of Helmholtz’s Die Lehre von den Tonempfindungen, for determining the relative
distance between two distinct pitches. Cent is defined as the 1200th root of 2, or 21~1200 , yielding the
ratio 1 1.000577789506555. It follows that there are 1200 cents to an octave ( 1.0005781200 2). Cents
are represented by the “” sign. The equation for calculating the cent value of a given frequency ratio is
{1200 log 2 R = }, or {(1200 / log10 2) log 10 R = }. The reverse operation is carried out by the
formula {2( / 1200) }. A hundred cents makes an “equal tempered semitone” (one degree of 12-tone Equal
Temperament), hence the origin of the term. (Accompanying endnotes)
179
In order to overcome aforesaid intonation problems induced by mainstream Turkish and
Arabic-Persian tone-systems, which fail to comprise the minimal amount of crucial intervals
to satisfactorily and wholesomely represent Makam music across popular instrumental
transpositions, the authors shall present herewith a novel 36-tone hybrid solution.
2. Quest For The Ideal Medium-Resolution Tuning
There is a general tendency in Turkey and the Levant, to divide the octave practically
into 53 logarithmically equal parts (i.e., 1200~53 22.64151 ) via the mathematical
operation shown in Formula 2.1 below:
Formula 2.1 (53 pitches apart by relative frequency values of the “Holderian comma”)
f 2
f

(n{1,2,3,... 53} /53)
º
53
f exp
2

or
n{1,2,3,... 53}
(2.1a)
or else,
n1, 2, 3, ...53~53 ln
(2.1b)
2 ,
(2.1c)
which is a voluminous resolution that embodies AEU with maximum 1 cent absolute error
at any degree [58]. This “Holderian comma system” helps musicians educated according
to AEU theory to conceptualize and communicate the positioning of at least two nonsystematized middle seconds (1200~53 6 135.85  & 1200~53 7 158.49 ), by counting
comma step deviations from certain pivotal notes in a melody [46].
It goes without saying, that such microtones are not ordinarily expressed in Turkish Art
music notation vii. Hence, musicians say, for example, that a certain pitch (perde) is to be
sounded one or two or three commas higher or lower than written [46].
It is also significant to emphasize at this point, that makam-oriented Turkish music
computer programs viii also utilize the Holderian comma resolution for true-to-the-original
digital playback of traditional music scores [64].
However, there happens to be a fundamental setback with 53-tone Equal Temperament:
There is not a Turkish instrument which is known to implement it faithfully or wholly. No
Kanun, Tanbur, Cümbüş, or Bağlama to date is prepared to embrace Holderian commas
in exactness or thoroughly.
Instead, most Tanburs utilize an arbitrary array of about 34 frets (destans) from perde
yegah (RE) to neva (Re), any of which can be moved around by performers on demand.
On the other hand, Kanun-makers haphazardly affix the half-tone mandal at the equal
semitone (100 cents) by referring to electronic tuners, and visually partition the space
between this mandal and the nut into 6 equal parts – arriving, to all intents and purposes,
at 72-tone Equal Temperament ix [59,60].
vii However, Folk music scores do utilize comma numbers above ordinary sharps and flats to indicate
the desired degree of 53-tone Equal Temperament.
viii “Notist” by Uğur Keçecioğlu, “Nota” by Ömer Tulgan, and “Mus2Okur - Turkish Music Multimedia
Encyclopedia” by M. Kemal Karaosmanoğlu & Data-Soft team of developers.
ix Actually, only a bulky subset of 72-tone Equal Tempermant (tET) can be found on quotidian Turkish
Kanuns, since not all degrees of 72-tET are available due to a general lack of need. Because of this setup,
certain transpositions are not possible; viz., the Kanun can only accompany an ensemble adjusted to one
180
Although 53-tET and 72-tET are very agreeable replacements for extended Just Intonation, either of which maintains approximated neutral or middle second savors sought after
by executants, they are unwieldy temperaments – an observation compounded by the fact
that only ad hoc subsets of these are applied to Turkish music instruments.
One can therefore ask, whether +50 tones per octave is really necessary as the definitive
groundwork of a music theory, or when performing on an instrument such as the Kanun
in an ensemble... In other words, is it fair to expect the Tanbur or Kanun to precisely
imitate by discrete static quanta the continuously intoned voice of Singers, Ud, Kemençe,
or Ney whose pitches are de facto not strictly bound to any particular theoretical grid?
We may ask this question all the more, since the first author had implemented a 79tone tuning on a unique Kanun in order to deliver a conclusive answer to the quest for
the least voluminous fixed-pitch resolution required to faithfully express Makam music in
every detail over all degrees of transposition [60].
Therefore, not only 53-tET and 72-tET do not anyway possess enough detail to fully
represent the free-pitch capabilities of versatile Middle Eastern instruments, the authors
further believe that, there should be room for subtle inflexions in a fixed-pitch tuning
strategy at any case – insofar as the stabilized set of perdeler can serve to represent a given
makam without altogether sounding out of place.
Accordingly, not only should the sought-after master tuning support bare instances of
steady-state intonation without doing injustice to the makam, but the chosen framework
ought not be cumbersome for the non-challenging notation and execution of temporally
standstill (but even so melodically functional) microtones.
41 tones to the octave thence appears to be the upper limit for a medium-resolution fixedpitch tuning strategy for Turkish Art music – since it is the lowest possible equal division to
feature a cycle of almost pure fifths (1200~41 24 702.44 ) x, while embodying at least
one minor second (1200~41 6 175.61 ) and one neutral second interval (1200~41 5
146.3415 ) critical to the essence of the Makam music genre [59].
However, this 41-tone resolution is still arguably a highly crowded selection. On the
other hand, the lower limit for a medium-sized template can be practically determined at
24-tones to the octave. Whereas subtle nuances must inevitably be sacrificed due to the
lessening of pitches, such is, unavoidably, the price to pay for a simple theoretical cast with
a gentle learning-curve.
With this in mind, the first author had proposed an alternative 24-tone irregular tuning
to AEU named Yarman-24 xi [41], which embraces characteristic neutral seconds at crucial
locations for Saba, Uşşak, Hüseyni, Hüzzam, Karcığar, etc..., while still relying on the same
palette of accidentals as AEU [63]. Given enough room for pitch inflexions (1 Holderian
comma berth per pitch for instance), it suffices to reasonably explain all makams over
of the more mainstream concert pitches (Ahenk or Akort). Sometimes, the equal semitones are observed
to be asymmetrically divided into 7 parts in the lower registers owing to available space (which yields
84-tET), and into 5 parts in the upper registers because of limited space (which yields 60-tET), at the
expense of pure octave complements for intra-semitonal microtones.
x Just 0.484  greater than the actual pure fifth (3:2) equal to {1200 ~ log 2 log 1.5} = 701.955 
10
10
xi As currently listed in the SCALA Program’s (accompanying endnote) Scale Archive authored by
Manuel op de Coul (YA24 notation in SCALA). It is not foreign, under the discipline of constructing
tunings & temperaments, to have scales named after their creator, given that there are thousands of them
to reckon in the literature, and that this procedure facilitates their cataloguing.
181
at least a single chosen Ahenk (or Akort). When pitted against pitch measurements from
masters of Turkish Art music in our previous article [13], Yarman-24 scored almost as high
as Mus2Okur spearheaded by the second author, which employs the voluminous Holderian
comma resolution.
Several other variants were advanced after Yarman-24a (christened “b”, “c”, “d ”), all
of which can likewise be notated using exactly the same arsenal of accidentals as AEU.
In particular, Yarman-24c has been applied by the first author to the neck of his bowed
Tanbur xii, and was furthermore implemented on the fretboard of a guitar belonging to
Tolgahan Çoğulu, as well as on TouchKeys “Capacitive Multi-Touch Sensing on a Physical
Keyboard” technology by Andrew McPherson [18,39,65]. Especially, the bowed Tanbur
and the TouchKeys keyboard can let a musician become quite liberal with pitch inflexions
using the Yarman-24 layout.
Nevertheless, enforcing the restricted usage of only AEU accidentals leads to an irregular
mapping of notes, which results in the sanctioning of notational inconsistencies for available
transpositions (viz., a given number of steps do not always correspond to the same type
of interval). Besides, not being able to transpose the body of makamlar over to at least
the main Ahenks without a frequency shift of the whole keyboard, or altering the tuning
of strings, can become a performance hindrance for certain settings.
On those accounts, a much less voluminous 36-tone alternative compared to 72-tET,
53-tET, and 41-tET shall be presented herein shortly, which features a mathematically
rigorous, yet straightforward, fixed-pitch tuning strategy to the problem of adequately
sounding and notating of essential Turkish makam genera throughout mainstream transpositions.
3. A 36-tone Replacement in place of Arel-Ezgi-Uzdilek
The so-referred Yarman-36 makam tone-system proposed in this paper comprises 36
tones locatable just by ear, via counting exact 0, 1, and 2 (and optionally 3) beats per
second when listening to given octave, fifth and third intervals as outlined in Fig. 2,
starting from an algebraically attained reference frequency for A at 438.41 Hertz, very
near the international standard A=440 Hz.
Said tuning cast is based on a twelve-by-twelve triplex structure of exclusively tailored
Modified Meantone Baroque Temperaments (each completing a fifths circle at the 12th
step), with aurally pleasant shades of key-colors supporting polyphony endeavours in line
with Western common-practice harmony and chordal modulation, while also accounting for
hitherto omitted pitches in Uşşak, Saba, Hüzzam, etc... – in contradistinction to the praxismismatched Arel-Ezgi-Uzdilek (AEU) music theory in force – at popular transpositions that
correspond to habitual Ahenkler (i.e., 12 or more possible concert pitches, with Bolahenk
at Re=440 Hz as the accepted default) as shown in Table 1 and Table 2.
The reason for choosing 438.41 Hertz as the reference frequency for note A is to assure
that the fifths cycle in Layer I is completed using only fifths with beat rates of 0, 1, and
2 per second throughout. While not at all a prerequisite of the Yarman-36a cast, said
reference frequency can be calculated by Formula 3.1 presented further below.
xii Ordinarily, open strings of Tanbur correspond to Bolahenk Akort, with perde yegah (melody-making
open string) at A (Re=220 Hz in Turkish parlance) according to international pitch; but the instrument
in question has been successfully tuned a perfect fourth sharper to Mansur Akort with perde yegah at D.
Layer I
! >
-1/8 PC
!
-1/8 PC
-2
beats per sec.
>
-1
>
>
>
...
>
-1
>
...
-2
-1
>
>
X>
>
-2
+1/16 PC
>
-1
X>
+1
-1
V>
V>
pure
+1/16 PC
X>
+1
V>
-2
X>
X>
0
V>
(5:3)
(PC: Refers to the Pythagorean comma of 531441:524288)
Layer II
+1/8 SC
! > rrrrrr ° >
-2
+3 (5:6)
! °>
-1/8 PC
-1
°>
°>
-1/8 PC
°>
...
-1
...
°>
°>
-1
-1
°>
¤>
°>
-1
-2
¤>
°>
¤>
°>
-1
pure
0
°>
+3/22 SC
¤>
V>
pure
0
V>
¤>
V>
-1/8 PC
V>
-1
(SC: Refers to the Syntonic comma of 81:80 – The jump is made from note G of Layer I)
Layer III
! > rrrrrr ± >
-2
(11:9)
! ±>
-1/8 PC
-1
±>
±>
-1/8 PC
±>
...
-1
...
±>
±>
-1
-1
±>
¢>
±>
-2
-1
¢>
±>
¢>
±>
-1
pure
0
¢>
+3/22 SC
¢>
pure
0
V>
¢>
¢>
¢>
-1/8 PC
-1
¢>
Figure 2: Tuning recipe for Yarman-36a tone-system via 0, 1, 2 (and optionally 3)
beat counts per second from octave, fifth and third intervals; followed
by 1/8 comma Temperament Ordinaire approximation guidelines.
deg.
0
0
49
49
1
nim zengule
31
80
2
nerm zengule
18
98
3
zengule
56
29
Rast
Dügah
16
dik rast
dik dügah
52
nim zengule
kürdi
31
nerm zengule
dik kürdi
22
zengule
nerm segah
49
dik zengule
segah
29
Rast
Dügah
Buselik
14
dik rast
dik dügah
dik buselik
57
nim zengule
kürdi
nerm çargah
31
nerm zengule
dik kürdi
Çargah
17
zengule
nerm segah
dik çargah
49
dik zengule
segah
nim hicaz
29
Dügah
Buselik
nerm hicaz
14
dik dügah
dik buselik
hicaz
60
kürdi
nerm çargah
dik hicaz
29
Rast
dik kürdi
Çargah
Neva
16
dik rast
nerm segah
dik çargah
dik neva
49
nim zengule
segah
nim hicaz
nim hisar
31
nerm zengule
Buselik
nerm hicaz
nerm hisar
22
zengule
dik buselik
hicaz
hisar
51
dik zengule
nerm çargah
dik hicaz
dik hisar
29
Rast
Dügah
Çargah
Neva
Hüseyni
14
dik rast
dik dügah
dik çargah
dik neva
dik hüseyni
54
nim zengule
kürdi
nim hicaz
nim hisar
acem
31
nerm zengule
dik kürdi
nerm hicaz
nerm hisar
dik acem
20
zengule
nerm segah
hicaz
hisar
nerm eviç
49
dik zengule
segah
dik hicaz
dik hisar
eviç
29
Dügah
Buselik
Neva
Hüseyni
Mahur
14
dik dügah
dik buselik
dik neva
dik hüseyni
dik mahur
60
kürdi
nerm çargah
nim hisar
acem
nerm gerdaniye
31
dik kürdi
Çargah
nerm hisar
dik acem
Gerdaniye
14
153 182 199 251 282 304 352 382 396 453 484 501 550 580 594 654 683 700 749 780 802 853 882 897 951 982 1002 1051 1080 1095 1155 1186 1200
4
dik zengule
BOLAHENK
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Table 1: Table of transpositions in main Ahenks via Yarman-36a tuning with corresponding microtonal notation.
<DUPDQPDNDPWRQHV\VWHP
cent
0
dik rast
DAVUD
33
34
35
36
! ! : |! ~! ;! ! ! : |! ! :
|! ~! ;! ! ! : |! ~! ;! ! ! : |! ! : |! ~! ;! ! ! : |! ~! ;!
int.
Rast
MANSUR
5
SÜPÜRDE
KIZ
184
Deg. & Note
0
48.963
80.006
97.641
153.152
182.378
198.747
250.591
281.923
303.638
352.336
381.641
396.078
452.588
483.954
501.356
550.227
579.633
594.119
654.228
683.403
699.744
748.768
779.85
801.683
853.083
882.343
896.757
950.633
981.999
1001.88
1050.682
1080.048
1094.514
1154.543
1185.909
1200
Yarman-36a
cent values
261.1692
268.661
273.522
276.3223
285.326
290.1836
292.9404
301.8455
307.3581
311.2376
320.1167
325.5816
328.308
339.2012
345.4028
348.8922
358.8813
365.0293
368.0965
381.1013
387.5782
391.2538
402.4915
409.783
414.9835
427.489
434.7755
438.4105
452.2682
460.5371
465.8562
479.1751
487.3724
491.4619
508.8018
518.1043
522.3384
1/8 comma
Frequencies Temperament
approximation
0
44.475
77.76
101.955
149.363
182.648
198.045
248.386
281.671
302.933
347.408
380.693
396.09
449.363
482.648
500.978
545.453
578.738
598.534
650.341
683.626
699.023
743.498
776.783
803.91
848.386
881.671
897.067
947.408
980.693
1001.955
1046.43
1079.715
1095.112
1151.318
1184.603
1200
0: C
1: Cx
2: CÕ
3: C#/D¬
4: Dă
5: D¬
6: D
7: Dx
8: DÕ
9: D#/E¬
10: Eă
11: E¬
12: E
13: Ex
14: F¬
15: F
16: Fx
17: FÕ
18: F#/G¬
19: Gă
20: G¬
21: G
22: Gx
23: GÕ
24: G#/A¬
25: Aă
26: A¬
27: A
28: Ax
29: AÕ
30: A#/B¬
31: Bă
32: B¬
33: B
34: Bx
35: c¬
36: c
0
4.487
2.246
-4.314
3.789
-0.27
0.702
2.205
0.252
0.705
4.928
0.948
-0.012
3.225
1.306
0.378
4.774
0.895
-4.415
3.887
-0.222
0.721
5.27
3.067
-2.227
4.697
0.672
-0.311
3.225
1.306
-0.075
4.251
0.332
-0.598
3.225
1.306
0
Difference
in cents
KABA (PES) RAST
kaba dik rast
kaba nim zengule (şuri)
kaba nerm zengule
kaba zengule
kaba dik zengule
KABA DÜGAH
kaba dik dügah
kaba kürdi
kaba dik kürdi (nihavend)
kaba nerm segah (uşşak)
kaba segah
KABA BUSELİK
kaba dik buselik
kaba nerm çargah
KABA ÇARGAH
kaba dik çargah
kaba nim hicaz
kaba nerm hicaz (uzzal)
kaba hicaz (saba)
kaba dik hicaz
YEGAH
dik yegah
kaba nim hisar
kaba nerm hisar (beyati)
kaba hisar (hüzzam)
kaba dik hisar
HÜSEYNİAŞİRAN
dik hüseyniaşiran
acemaşiran
dik acemaşiran
nerm ırak
ırak
GEVAŞT (REHAVİ)
dik gevaşt
nerm rast
RAST
Bolahenk = +6 deg.
Davud = +12 deg.
Mansur = -15 deg.
Kız = -9 deg.
SÜPÜRDE perde
names of 1st octave
RAST
dik rast
nim zengule (şuri)
nerm zengule
zengule
dik zengule
DÜGAH
dik dügah
kürdi
dik kürdi (nihavend)
nerm segah (uşşak)
segah
BUSELİK
dik buselik
nerm çargah
ÇARGAH
dik çargah
nim hicaz
nerm hicaz (uzzal)
hicaz (saba)
dik hicaz
NEVA
dik neva
nim hisar
nerm hisar (beyati)
hisar (hüzzam)
dik hisar
HÜSEYNİ
dik hüseyni
acem
dik acem
nerm eviç (nevruz)
eviç
MAHUR
dik mahur
nerm gerdaniye
GERDANİYE
Bolahenk = +6 deg.
Davud = +12 deg.
Mansur = -15 deg.
Kız = -9 deg.
SÜPÜRDE perde
names of 2nd octave
GERDANİYE
dik gerdaniye
nim şehnaz
nerm şehnaz
şehnaz
dik şehnaz
MUHAYYER
dik muhayyer
sünbüle
dik sünbüle
tiz nerm segah (uşşak)
tiz segah
TİZ BUSELİK
tiz dik buselik
tiz nerm çargah
TİZ ÇARGAH
tiz dik çargah
tiz nim hicaz
tiz nerm hicaz (uzzal)
tiz hicaz (saba)
tiz dik hicaz
TİZ NEVA
tiz dik neva
tiz nim hisar
tiz nerm hisar (beyati)
tiz hisar (hüzzam)
tiz dik hisar
TİZ HÜSEYNİ
tiz dik hüzeyni
tiz acem
tiz dik acem
tiz nerm eviç (nevruz)
tiz eviç
TİZ MAHUR
tiz dik mahur
tiz nerm gerdaniye
TİZ GERDANİYE
Bolahenk = +6 deg.
Davud = +12 deg.
Mansur = -15 deg.
Kız = -9 deg.
SÜPÜRDE perde
names of 3rd octave
Highest absolute :
5.2703
Average absolute :
2.0902
Root mean square : 2.7276
Total absolute :
75.2472
Table 2: Table of pitch data for Yarman-36a tuning.
185
Formula 3.1 (calculation of the specific reference frequency "f" for note LA via the
elimination of the fifth beat rate between G#–Eb of Layer I in Fig. 2 )
2
f
α β γ δ ε ζ 8 3
f
a b c d e~16
0
(3.1a)
Sol®
e
3
f
a
b c d 8
2 f a b c d
f
a
f
a
(3.1b)
d
c
b
Do®
a
3
b c 4
2 f a b c
(3.1c)
Fa®
3
b 4
2 f a b
(3.1d)
Si
3
f
a 4
2 f a
(3.1e)
Mi
3
f2
f
(3.1f)
f2
3 f
(3.1g)
2
α
β
γ
δ
ε
LA (f )
ζ
2
Re
2
f
α 1
3 f α
(3.1h)
Sol
2
f
2
f
α
β 0.5
3 f α β
(3.1i)
Do
3
α
f
β γ 0.5
β γ
(3.1j)
β γ δ 0.25
β γ δ
(3.1k)
β γ δ ε 0.25
β δ γ ε
(3.1l)
α
Fa
2
f
3
α
f
α
Si¬
Mi¬
2
f
3
α
f
α
186
Formula 3.1 – continued – (calculation of the specific reference frequency "f" for
note LA via the elimination of the fifth beat rate between G#–Eb of Layer I in Fig. 2 )
(3.1m)
where Formula 3.1a, via the expansion of all its associated terms, results in the equation
shown in 3.1m, whose outcome is f = 3135950 / 7153 , which makes 438.41046 Hz for
note A. This is simply to assure that the fifth between G# and Eb comes out pure at the
end. One can, at any case, optionally disregard such a route by choosing the international
standard A=440 Hz. Doing so does not conceptually affect the Yarman-36a tuning scheme
in the least. On the other hand, a lower A is authentic not only for Western Classical music,
but also for Ottoman-era music.
Whereas, the first author had formulated two more variants after his initial Yarman-36a
(christened “b” and “c”), both of which are constructed as triple cascading quasi-equally
tempered 12 tones apiece, only the original Yarman-36a will be undertaken in this paper.
Regardless, any of the Yarman-36 variants can be implemented on a Kanun, Tanbur,
Cümbüş, Bağlama, or mapped to a tripartite Halberstadt keyboard layout; and all of them
readily feature approximations for both the comma nuances and one kind of critical neutral
second peculiar to the essence of the genre xiii – that are comprised in whole by neither
AEU nor the Arabic/Persian 24 tone cast.
To rephrase, Yarman-36a is a triple-layered “Baroque-style” 1/8 comma Modified Meantone cyclic tuning, capable of decently expressing makams over Süpürde, Bolahenk, Davud,
Mansur and Kız Ahenks, while also making possible the elegant and authentic sonorities of
European music, alongside several exotic microtonal chords of modern xenharmony. The
proposed Yarman-36a makam tone-system illustrated in Table 1 and Table 2 not only furnishes crucial commatic, neutral, and sesquitone (augmented second) intervals demanded
by traditionalist executants of the Middle East altogether in a single package, it further
facilitates Western-oriented musicians’ understanding of makamlar through the suitable
xiii Differences between the Yarman-36 a, b, c variants are minute – that is to say, a musician can swap
one for the other with only slight (maximum 9 cents per degree) intervallic deformity. Such divergence
ought not arouse significant aural discomfort since a few cents mistuning of intervals is observed to be
indiscernable in traditional ensembles or orchestras composed of complex timbres. Besides, the “b” and
“c” variants are solely the product of mathematical perfectionism as one searches for intervallic regularity.
Nevertheless, to summarize: Yarman-36a features pitch relations yielded by selective 0, 1, and 2 integer
beats per second based on a dedicated reference frequency for A at 438.41 Hz, with 2~1 as octave; Yarman36b thrice collates in identical triplex fashion equally spaced twelve pitches per layer with 441~220 (1204
cents) as the octave; and the almost entirely rational Yarman-36c comprises mostly pure fifths in like vein
as version “a”, with again 441~220 as octave.
Yarman-36a, subject to further elaboration hereunder, is the easiest to implement acoustically and
without electronic aid. Yarman-36b is the closest tuning to 12-tone Equal Temperament with only 4 cents
absolute difference at any degree, while possibly being the hardest to tune by ear – making it perhaps an
ideal regular Temperament model when discussing theory on paper. Yarman-36c flaunts proportionally
beating chords that ought to please the listener due to an abundance of rational pitches, rendering it the
obvious choice for digitally pedantic expositions on extended Just Intonation. No further mention of the
“b” and “c” variants are required at this point.
187
employment of enharmonically equivalent (i.e. respellable) sharps & flats at simple key
signatures.
A consistent microtonal staff notation tailored to express Yarman-36 makam tonesystem maintains all of the accidental symbols of AEU, with the addition of merely a
sharp & flat pair more for degrees 2, 8, 17, 23 and 29. This specialty makes it quite easy to
convert from AEU notation to the Yarman-36 makam tone-system, as can be seen in Fig.
3. The flexibility of intervals depending on the transposition means that, the accidentals
occupy regions on the whole-tone continuum, as illustrated in Fig. 3 and Table 3.
Figure 3. Conversion scheme from Arel-Ezgi Uzdilek, where accidentals of
Yarman-36a occupy regions on the 9 Holderian commas wide whole-tone
continuum, and where only one extra sharp & flat pair is needed.
Table 3. Extent of common microtonal accidentals from all natural notes
in the Yarman-36a tuning.
x
Õ
#
ă
¬
7
48.9-60  78.3-91.4  92.8-105.5  -43.7 to -48.8  -14.1 to -17.4  197-203.1 
An alternative palette of accidentals that are more amiable to the Persian sori and koron
symbology is also possible, and perhaps more preferable for international standardization
concerns. They are given in Fig. 4. The only change compared to Figure 3 is regarding
the “lesser (¸) sharp” and the “greater (¹) sharp”.
Figure 4. Alternative accidentals for notating Yarman-36 that are more amiable
to the Persian sori (1/4-tone sharp) and koron (1/4-tone flat) symbology.
188
4. Analysis and Conclusions
Tetrachords and pentachords of Turkish Art music can henceforward be re-defined
using the Yarman-36a cast. A catalogue of complete genera are attempted in Figs. 5 &
6 throughout Süpürde, Bolahenk, Davud, Mansur and Kız Ahenks in the following pages.
Once they have been transcribed thus, it is possible to conjoin them in the construction
of characteristic makam scales. Due to exhaustion of space in this article, such work is
postponed to a future study.
We can nevertheless engage in a comparison of select genera with their AEU counterparts in Table 4 below, by referring each to pentachordal subsets of histogram peaks
achieved from recordings by master performers [13,15]. The peaks were collated from 128
pieces in 9 makam categories and can be readily matched to 8 genera in the table. Also,
since Uşşak and Hüseyni are ordinarily identified with the same intervallic structure in
AEU, the average of their respective peaks are taken.
Table 4. Comparison of genera in AEU and Yarman-36a with pitch measurements
Genus
Rast tetrachord
(Rast pentachord)
Uşşak tetrachord
AEU (Hc) Measr.(Hc) Measr.()
9
9.17
207.6
+8
+7.47
376.7
+5
+5.26
495.8
(+9)
(+9.12)
(702.3
8
+5
+9
(+9)
6.32
+6.16
+9.36
(+9.36)
143.1
282.6
494.5
(706.4)
9
+4
+9
(+9)
9.38
+3.52
+9.17
(+8.88)
212.4
292.1
499.7
(700.8)
4
+9
+9
(+9)
5.26
+7.45
+9.19
(+9.2)
119.1
287.8
495.9
(704.2)
5
+12
+5
(+9)
4.65
+12.16
+4.98
(+9.3)
105.3
380.6
493.4
(704)
5
+9
+8
(+9)
4.68
+9.4
+9.03
(+7.58)
106
318.8
523.3
(694.9)
(Saba1 Pentachord)
8
+5
+5
(+13)
7.61
+5.18
+5.91
(+12.44)
172.3
289.6
423.4
(705.1)
Hüzzam Pentachord
5
+9
+5
+12
4.99
+9.18
+6.28
+10.52
113
320.8
463
701.2
(Hüseyni2 pentachord)
Buselik tetrachord
(Buselik pentachord)
Kürdi* tetrachord
*Kürdilihicazkar
(Kürdi* pentachord)
Hicaz tetrachord
(Hümayun pentachord)
Segah tetrachord
(Segah pentachord)
Saba (dim.) Tetrachord
AEU ()
203.9
384.4
498
(702)
Average:
180.4
294.1
498
(702)
Average:
203.9
294.1
498
(702)
Average:
90.2
294.1
498
(702)
Average:
113.7
384.4
498
(702)
Average:
113.7
317.6
498
(702)
Average:
180.4
294.1
407.8
(702)
Average:
113.7
317.6
431.3
702
Average:
Grand Avg.:
Diff.
-3.7
7.7
2.2
-0.3
3.48
37.3
11.5
3.5
-4.4
14.18
-8.5
2
-1.7
1.2
3.35
-28.9
6.3
2.1
-2.2
9.88
8.4
3.8
4.6
-2
4.7
7.7
-1.2
-25.3
7.1
10.33
8.1
4.5
-15.6
-3.1
7.83
0.7
-3.2
-31.7
0.8
9.1
7.85625
YA36 mean ()
197.8
382
501.1
(698.9)
Average:
154.1
303.2
501
699.6
Average:
200.8
287.6
501.2
(699.8)
Average:
103
303.1
501.1
(698.9)
Average:
105.3
384.7
501.2
(699.8)
Average:
119.1
316.9
500.5
(700.1)
Average:
184.2
287.6
403.8
(699.8)
Average:
119.1
316.9
470.8
700
Average:
Grand Average:
Diff.
-9.8
5.3
5.3
(-3.4)
5.95
11
20.6
6.5
(-6.8)
11.23
-11.6
-4.5
1.5
-1
4.65
-16.1
15.3
5.2
-5.3
10.48
0
4.1
7.8
-4.2
4.03
13.1
-1.9
-22.8
5.2
10.75
11.9
-2
-19.6
-5.3
9.7
6.1
-3.9
7.8
-1.2
4.75
7.6925
Yarman-36 makam tetrachords
SÜPÜRDE
Rast
! < > °> <
198.7
119.7
182.9
Arabi
Rast
! < ±> > <
149
Mahur
198.7
153.6
! < > > <
198.7
105.3
197.3
Kürdili
Hicazkar
! < V> V> <
97.6
197.7
206
Kürdi
! < > ±> <
153.6
! < V> ¤> <
104.9
120.1
276
! < ¢> ± > <
149.5
153.6
197.9
Segah
! °< > > °<
119.7
182.6
198.4
Uzzal /
Çargah
! < ±> > V<
152.9
105.1
242.5
Suzinak
149.5
197.3
154.1
! < ±> °> <
153.3
120
227
DAVUD
< X> ¤> <
198
116.9
185.7
< ¢> X> <
148
198
154.6
< > X> <
< X> X> <
< V> > <
105.3
197
198.4
197.3
105.6
198
104.9
198.4
197.7
183.6
197
120.1
149
Garip
< ¢> > <
! < °> > <
198.4
219.4
198.4
Hicaz
197.3
120.1
183.6
< °> > <
182.9
198.4
119.7
Uşşak
< > ¤> <
! < ´> > <
83.2
Neva
BOLAHENK
87.9
197
215.8
< ¤> > <
< > ¢> <
197
154.1
149.5
< > ¤> <
105.3
116.9
278.5
< ¢> ¢ > <
148
154.1
198.5
¤< > > °<
120.1
183.3
< ±> > <
197
153.3
105.5
241.4
< ±> ¤> <
153.9
118.7
229.3
198
95.1
207.6
< > > <
X< °> > <
89.3
197.8
213.4
X< ¤> > <
185.7
197.8
116.9
< > ¢> X<
197.8
154.6
148
X< > ¤> <
105.6
112.5
282.3
< ¢> ¢ > X <
143.9
154.6
201.9
¤< > > ¤<
116.9
185.5
197.8
< ±> X> <
153.9
101.1
< ¢> ¤> <
247
154.4
114.2
233
< > °> <
MANSUR
197
120
< ±> > <
183.3
149.3
197
153.9
< > > <
197
105.5
197.8
< > ¤> <
KIZ
197.8
118.7
185.5
< ¢> > <
149.8
197.8
154.4
< > X> <
197.8
101.1
203.1
< V> V> <
< V> > <
85.2
91.4
101.9
198.1
200.2
< ´> > <
198.7
< °> > <
218
183.3
198.7
< > ±> <
120
198.7
153.9
149.3
< V> ¤> <
105.1
118.7
278.1
< ¢> ±> <
149.8
153.9
198.3
105.1
198.7
198.1
< °> > <
197.3
212.8
< ¤> > <
185.5
197.3
118.7
< > ¢> <
197.3
154.4
149.8
< > ¤> <
105.5
114.2
281.9
< ¢> ¢> <
145.5
154.4
201.6
°< > > °<
¤< > > ¤<
153.2
105.3
242.9
153.6
105.6
241.8
120
182.9
198.7
118.7
183.6
197.3
< ±> > <
< ±> X> <
153.6
120.1
227.3
154.1
116.9
229.6
< ±> ¤> <
< ¢> ¤> <
Figure 5: Notation of Yarman-36a makam tetrachords in main Ahenks
with consecutive intervals in cents
Yarman-36 makam pentachords
< > ¤> > <
< X> ¤> > <
< > °> > <
! < >± > > <
198.4 153.6
149 198.7
< > ¢> > <
< > ¢> X> <
< > ±> > <
! <> > > <
< > X> > <
< X> X> > <
< > > > <
SÜPÜRDE
Rast
! < > °> > <
198.7 119.7
182.9 198.4
Arabi
Rast
Mahur
198.7 105.3
197.3 198.4
Pençgah
Nikriz
Kürdili
Hicazkar
198 95.1
207.6 197.8
198.7 153.9
149.3 197
197 105.5
197.8 198.7
198.7 198
182.9 120.1
197.3 200.2
183.6 116.9
< > °> ¤> <
! < > V> ¤> <
198.7 276
104.9 120.1
< > > ¤> <
197.3 278.5
105.3 116.9
< X> > ¤> <
< > V> ¤> <
! <V> V> > <
< V> > > <
< > > > <
105.3 197
198.4 197.8
< V> V> > <
101.9 198.1
200.2 198.7
< > > > <
< X> > > <
< > V> > <
! < > °> > <
< X> °> > <
X< X> °> > X<
< > °> > <
! <°> > > <
< ¤> > > <
183.6 197
120.1 197.8
X< ¤> > > X<
! < > >±> <
< > > ¢> <
X< > > ¢> X<
! < > V> > <
182.9 198.4
119.7 197
Hüseyni2
197.3 105.6
198 197
197.8 154.6
148 198
197 120
183.3 198.7
< X> ¤> ¤> <
197.3 215.8
87.9 197
Hüseyni1
197 154.1
149.5 197.3
198 116.9
185.7 197.8
MANSUR
< > ¤> ¤> <
198.7 197.7
104.9 198.4
Buselik
197.3 120.1
183.6 197
DAVUD
! < > °> ¤> <
97.6 197.7
206 198.4
Nihavend
BOLAHENK
197 149
198.4 153.6
104.9 198.4
197.7 197
197.3 198.4
105.3 197
198 213.4
89.3 197.8
197.8 149.5
197 154.1
198 202.1
185.7 112.5
198 282.3
105.6 112.5
198
197
105.6 197.8
207.6 212.2
80.7 203.1
185.7 197.8
116.9 203.1
203.1 148
197.8 154.6
197 200
183.3 118.7
197 278.1
105.1 118.7
197 198.1
105.1 198.7
< > ¤> > <
KIZ
197.8 118.7
185.5 197.3
< > ¢> > <
197.3 154.4
149.8 197.8
< > X> > <
197.8 101.1
203.1 197.3
< > ¤> ¤> <
197.8 201.9
185.5 114.2
< > > ¤> <
197.8 281.9
105.5 114.2
< V> > > <
105.1 198.7
198.1 197.3
< > > > <
197.8 198.7
105.5 197.3
< X> °> > X<
197.8 212.8
91.4 197.3
203.1 213.7
84.7 198
183.3 198.7
120 197.3
185.5 197.3
118.7 198
< °> > > <
< > >±> <
197.3 149.3
198.7 153.9
< ¤> > > X<
X< > > ¢> <
198 149.8
197.3 154.4
Figure 6: Notation of Yarman-36a makam pentachords in main Ahenks
with consecutive intervals in cents
Hümayun
Uzzal /
Çargah
Suzinak
Saba 1
Saba 2
Saba
Zemzeme
Hüzzam
< > ¤> > <
105.3 116.9
278.5 197.8
105.6 112.5
282.3 203.1
105.1 118.7
278.1 197.3
105.5 114.2
281.9 198
! < > ¢>±> <
197 197.9
149.5 153.6
< > ¢> ¢> <
197.8 198.5
148 154.1
X< > ¢> ¢> X<
203.1 201.9
143.9 154.6
< > ¢>±> <
197.3 198.3
149.8 153.9
X< > ¢> ¢> <
! <± > > V > <
<±> > > <
<±> X> > <
<±> > > <
<±> X> > <
BOLAHENK
DAVUD
MANSUR
152.9 105.1
242.5 198.1
153.3 105.5
241.4 198.7
153.9 101.1
247 197.3
153.2 105.3
242.9 198.4
! <± > ° > > <
<±> ¤> > <
< ¢> ¤> > X<
<±> ¤> > <
198 201.6
145.5 154.4
153.6 105.6
241.8 197
< ¢> ¤> > <
153.9 118.7
229.3 197.3
154.4 114.2
233 198
! <°> °> V> <
182.9 110.2
102.3 302.6
< ¤> °> V> <
183.6 118.3
103.8 292.8
X< ¤> °> V> X<
185.7 119.5
102.5 295.8
183.3 111.7
105.9 298.4
185.5 121.3
102.4 290.5
! <± > > ± > <
242.5 149
152.9 153.6
<±> > ¢> <
241.4 149.5
153.3 154.1
X<±> > ¢> X<
247
148
153.9 154.6
<±> >±> <
242.9 149.3
153.2 153.9
X<±> > ¢> <
! <± > > V > <
242.5 197.7
152.9 104.9
<±> > > <
241.4 198.4
153.3 105.3
X<±> > > X<
247
197
153.9 105.6
<±> > V> <
242.9 198.1
153.2 105.1
X<±> > > <
! °< > >±> °<
¤< > > ±> ¤<
¤< > > ¢> ¤<
°< > >±> ¤<
¤< > > ¢> ¤<
!
¤< > > §> ¤<
¤< > X> §> ¤<
¤< > > §> ¤<
¤< > > §> ¤<
°< > > ¤> ¤<
120 285.2
198.7 95.7
120.1 153.9
197 229.3
118.7 287.3
197.3 96.4
116.9 154.4
197.8 233
114.2 288.2
198 99.7
153.6 120.1
227.3 197
KIZ
153.3 120
227 198.7
119.7 153.3
198.4 227
Irakeyn
< > ¤> > X<
! < V> ¤> > <
104.9 120.1
276 197
Garip
< V> ¤> > <
X< > ¤> > X<
SÜPÜRDE
< °> °> V> <
120 153.6
198.7 227.3
120.1 289.2
197 94.1
154.1 116.9
229.6 197.8
< ¤> °> V> X<
241.8 149.8
153.6 154.4
241.8 198.7
153.6 105.5
118.7 154.1
197.3 229.6
116.9 287.9
197.8 99.5
Figure 6 – continued: Notation of Yarman-36a makam pentachords in main
Ahenks with consecutive intervals in cents
Kürdi
< °> > > <
! <´ > > > ´ =
< ° > > >´ =
X< °> > > °=
4th
>
´
< > >´ =
4th
< °> > > °=
! °< > > °> °<
¤< > > °> ¤<
¤< > > ¤> ¤<
°< > > °> ¤<
¤< > > ¤> ¤<
! °< > > °> V=
¤< > > °> =
¤< > > ¤> =
4th
°< > > °> =
4th
¤< > > ¤> =
! °<¤> > > °<
¤< ¤> > > ¤<
¤< ¤> > X> ¤<
°< ¤> > > ¤<
¤< ¤> > X> ¤<
! ° < ¤> > >´ =
¤< ¤> > > °=
¤< ¤> > X> °=
°< ¤> > > °=
¤< ¤> > X> °=
!
¤< > > X> ¤<
¤< > X> X> ¤<
¤< > > > ¤<
¤< > > X> ¤<
°< > > > =
¤< > > X> =
¤< > X> X> =
¤< > > > =
4th
>
¤< > X> =
! <¤> > > V=
X< ¤> > > =
X< ¤> > X> =
< ¤> > > =
X< ¤> > X> =
4th
83.2 198.4
219.4 80.1
Segah
119.7 182.6
198.4 197.7
4th
dim.
Segah
119.7 182.6
198.4 119.5
Müstear
198 197
120.1 183.3
dim.
Müstear
198 197
120.1 85.2
Ferahnak
dim.
Ferahnak
(dim.)
Nişabur
< °> > > X<
! <´ > > > <
83.2 198.4
219.4 197
dim.
Kürdi
<´ > > > <
X< °> > > X<
SÜPÜRDE
!
°< > > > ¤<
120 197.3
198.7 183.6
4th
120 197.3
198.7 105.3
183.6 197
120.1 105.1
BOLAHENK
87.9 197
215.8 197.8
4th
87.9 197
215.8 85.2
120.1 183.3
197 200
4th
120.1 183.3
197 120
200.2 197.8
116.9 185.5
200.2 197.8
116.9 91.4
118.7 198
197.3 185.7
4th
118.7 198
197.3 105.6
185.7 197.8
116.9 105.5
DAVUD
89.3 197.8
213.4 203.1
4th
89.3 197.8
213.4 91.4
116.9 185.5
197.8 201.9
4th
116.9 185.5
197.8 118.7
202.1 203.1
112.5 184.3
202.1 203.1
112.5 84.7
114.2 207.6
198 180.3
4th
114.2 207.6
198 95.1
180.3 203.1
112.5 101.1
MANSUR
85.2 198.7
218 197.3
85.2 198.7
218 83.2
120 182.9
198.7 198
120 182.9
198.7 119.7
200 197.3
118.7 183.6
200 197.3
118.7 87.9
120.1 197.8
197 185.5
4th
120.1 197.8
197 105.5
185.5 197.3
118.7 105.3
KIZ
91.4 197.3
212.8 198
91.4 197.3
212.8 87.9
118.7 183.6
197.3 200.2
118.7 183.6
197.3 120.1
201.9 198
114.2 185.7
201.9 198
114.2 89.3
116.9 203.1
197.8 184.3
116.9 203.1
197.8 101.1
184.3 198
114.2 105.6
Figure 6 – continued: Notation of Yarman-36a makam pentachords in main
Ahenks with consecutive intervals in cents
193
We can right away see in Table 4 that, the cumulative errors of AEU are slightly
greater than the grand average of mean values across 5 Ahenks in Yarman-36a, despite the
fact that each tuning system can be improved further through better selection of pitches for
certain genera. For example, Yarman-36a could better approximate the 2nd steps of Uşşak,
and Kürdi as well as the 3rd step of Segah by the occasional employment of neighboring
pitches, and AEU might similarly correct for Kürdi as well as Segah. Notwithstanding,
such manipulations turn out to be more advantageous overall for Yarman-36a and are
therefore avoided.
Yet, this is about all AEU can achieve with its 24 tones, whereas our tuning proposition
fares much better against problematic genera such as Uşşak and Hüzzam, and also certain
known instances of Saba not immediately discernable from pitch measurements here –
which finely fit the broad diversity of tetrachord & pentachord definitions in the Yarman36 makam tone-system, with still more definitions possible.
To reflect the importance of each genera for the repertory, we can calculate a weighted
arithmetic mean by referring the outcomes of Table 4 to the percentage of pieces that
belong to corresponding makamlar. According to Timuçin Çevikoğlu [17], 45.2 % of the
total 23,592 pieces in 286 makams are composed in 1) Rast making up 1344 pieces, 2)
Uşşak & Hüseyni as well as Muhayyer & Bayati making up 1242 987 359 309 2897
pieces, 3) Buselik making up 346 pieces xiv, 4) Kürdilihicazkar making up 1275 pieces xv,
5) Hicaz making up 2359 pieces xvi, 6) Segah making up 601 pieces xvii, 7) Saba making up
431 pieces xviii, and 8) Hüzzam making up 1408 pieces xix. This data can now be used in
Table 5 to judge the real global distance of AEU and Yarman-36a from measurements:
Table 5. Global weighted average deviations, as referred to the repertory,
of AEU and Yarman-36a genera from pitch measurements.
Makam
RAST
UŞŞAK-HÜS.-MUH.-BEY.
BUSELİK
KÜRDİ (K.HİCAZKAR)
HİCAZ
SEGAH
SABA
HÜZZAM
Grand Average
Sum
Repertory % AEU Avg.
5.7 %
3.48
12.28 %
14.18
1.4666 %
3.35
5.4 %
9.88
9.999 %
4.7
2.55 %
10.33
1.83 %
7.83
5.97 %
9.1
7.86 cents
45.2 %
Weighted Avg. YA-36 Avg.
0.44
5.95
3.85
11.23
0.11
4.65
1.18
10.48
1.04
4.03
0.58
10.75
0.32
9.7
1.2
4.75
7.7 cents
8.72 cents
Weighted Avg.
0.75
3.05
0.15
1.25
0.89
0.61
0.39
0.63
7.72 cents
xiv Transposition of the genus in Nihavend makam is ignored, owing also to the controversy regarding
how Buselik is structurally distinct from it.
xv We cannot ascertain the contributing number of pieces in makam Kürdi from Çevikoğlu (that are
anyway outside the 72% comprising the foremost 20 makams), and also do not include its derivatives
such as suffixed makamlar like Muhayyer-Kürdi and Acem-Kürdi, which only feature the genus toward the
finalis.
xvi We cannot ascertain the contributing number of pieces in kindred Hümayun and Uzzal makams
from Çevikoğlu (that are anyway outside the 72% comprising the foremost 20 makams).
xvii Transposition of the genus in kindred Irak and Eviç makams are ignored.
xviii We do not include derivative composite modes such as Bestenigar and Çargah (that are anyway
outside the 72% comprising the foremost 20 makams).
xix Suzinak, which is a composite of Hüzzam & Rast, is ignored because it can belong to either category.
194
The calculations for Table 5 are done by multiplying the repertory percentages in
column 2 by either the AEU averages in column 3, or the Yarman-36a means in column
5, and then diving the resultant number by the repertorial sum 45.2 % to produce the
results in columns 4 and 6. These weighted averages columns are then cumulated to yield
the weighted average global outcomes – which are 8.72  overall deviation for AEU
and 7.72  overall deviation for Yarman-36a – which accounts for nearly half the
repertory.
As can be immediately noticed, the already poor performance of AEU at representing
Uşşak-Hüseyni-Muhayyer-Beyati and Hüzzam is worsened due to the abundant usage of
related genera in the repertory. In other words, characteristic and frequent occurence of
middle-second interval flavors lowers the score of AEU further. While not quite discernable
in the case of Saba here, the same situation is known to be true for Saba’s various auditions
too, where the second step may be flattened as much as a quarter-tone in descent to finalis.
All of these can be readily approximated by the available and additionally possible genera
in our tone-system.
In contrast to AEU, Yarman-36a accomodates the problematic genera fairly enough.
General intonational sacrifices such as detuned fifths, fourths, and thirds are compensated
thus. Subsequently, allowing for no more than a nominal 1 Holderian comma (1200~53
22.6 ) maximum pitch-bend flexibility lets Yarman-36a tone-system perform admirably
as a novel makam theory candidate.
Moreover, the Modified Meantone Temperament basis of Yarman-36a is agreeable with
the historical 9 steps to the whole-tone, 55 steps to the octave methodology of Europe,
known at the time of Georg Philipp Telemann and Leopold Mozart for approximating
1/6-comma tempered fifths tuning [42] – which was remarkably employed by Antoine de
Murat at the end of the 18th Century to explain minute alterations of pitch in Ottoman-era
Makam music to Westerners [3]. The slightly mellower 438.41 Hz reference frequency for
note A has historicity too under such a context, way before 440 Hz became the international
norm by the 20th Century [47].
Qualitatively speaking, Yarman-36 makam tone-system has greater explanatory power
in terms of
1) the potential to serve Western common-practice music via a 12-tone cyclic subset
easily tunable by ear and flaunting vibrant key-colors;
2) the capability to house quarter-tones, next to commatic nuances, to embrace a larger
geography;
3) its hybrid functionality in notating both Western and Middle Eastern musics using a
consistent array of accustomed accidentals that feature enharmonically equivalent sharps
and flats;
4) its success in fairly transposing Turkish makamlar over to five main Ahenks; and
5) its support for approximated Just Intonation polyphony, as well as provision for
substantial Xenharmonic resources.
Auditory-visual examples of some genera and chords can be discovered at the first
author’s website [62].
195
In conclusion, our hybrid tone-system proposal can appeal to not only Classical/
Contemporary Western musicians and Middle Eastern performers of traditional makam
instruments, but also to avant-garde composers searching for new microtonal expression
venues.
O U O
Acknowledgement The authors would like to extend their gratitude to Prof. Dr.
Tolga Yarman, Prof. Dr. Metin Arık, and Assist. Prof. Dr. Fatih Özaydın for their highly
valued input and assistance in the publication of this article.
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198
Ozan Uğraş Yarman Began Piano education at Kadıköy Conservatory.
Entered Gnessin Conservatory in 1992, Mimar Sinan Conservatory in 1993,
Brussels Royal Conservatory in 1994, and graduated as Pianist in 1997 with
the degree of "First Prize". In 1998, was accepted to the Postgraduate Composition Department of İstanbul University Conservatory. In 2001, earned
his Master’s degree. In 2002, was accepted to the Doctorate programme of
the Musicology Division of İstanbul Technical University Turkish Music Conservatory. In 2008, achieved his Doctorate degree. Was elevated to the rank
of Associate Professor in Musicology & Music Theories in 2011. Interested
in applying microtonal polyphony based on makamlar in his compositions.
M. Kemal Karaosmanoğlu Completed his Bachelor’s Degree in the Mathematics Department of İstanbul University, and Master’s Degree in System
Analysis Programme of İstanbul Technical University. After moving on to the
business world in 1984, founded Plekom Computer company. Since the 20032004 educational year, teaches Sound Programming, Musical Arithmetics,
Physics of Music, and Scale Theory courses as part-time lecturer under Audio
Design Programme of Yıldız Technical University. Between 2002-2011, has
assumed R&D management and sofware project leadership responsibilities in
Data-Soft Computer Ltd. and spearheaded the development of "Mus2okur Turkish Music Multimedia Encyclopedia".

197 - TWMS Journal of Applied and Engineering Mathematics

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